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A consumer research group is interested in testing an automobile manufacturer’s claim that a new economy model will travel at least 25 miles per gallon of gasoline ( H 0 : μ ≥ 25). a. With a .02 level of significance and a sample of 30 cars, what is the rejection rule based on the value of x ¯ for the test to determine whether the manufacturer’s claim should be rejected? Assume that σ is 3 miles per gallon. b. What is the probability of committing a Type II error if the actual mileage is 23 miles per gallon? c. What is the probability of committing a Type II error if the actual mileage is 24 miles per gallon? d. What is the probability of committing a Type II error if the actual mileage is 25.5 miles per gallon?

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STATISTICS F/BUSINESS+ECONOMICS-TE...

13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884

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STATISTICS F/BUSINESS+ECONOMICS-TE...

13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884
Chapter 9.6, Problem 49E
Textbook Problem
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A consumer research group is interested in testing an automobile manufacturer’s claim that a new economy model will travel at least 25 miles per gallon of gasoline (H0: μ ≥ 25).

  1. a. With a .02 level of significance and a sample of 30 cars, what is the rejection rule based on the value of x ¯ for the test to determine whether the manufacturer’s claim should be rejected? Assume that σ is 3 miles per gallon.
  2. b. What is the probability of committing a Type II error if the actual mileage is 23 miles per gallon?
  3. c. What is the probability of committing a Type II error if the actual mileage is 24 miles per gallon?
  4. d. What is the probability of committing a Type II error if the actual mileage is 25.5 miles per gallon?

a.

To determine

Find the rejection rule based on the value of x¯ for the test to determine whether the manufacturer’s claim should be rejected.

Explanation of Solution

Calculation:

The given information is that n=30, σ=3, and α=0.02.

The hypotheses are given below:

Null hypothesis:

H0:μ25

Alternative hypothesis:

Ha:μ<25

The rejection rule for lower tailed test using the critical value is,

If zzα, reject the null hypothesis

Critical value:

Software Procedure:

Step-by-step procedure to obtain the critical value using the MINITAB software:

  • Choose Graph > Probability Distribution Plot choose View Probability> OK.
  • From Distribution, choose ‘Normal’ distribution.
  • Click the Shaded Area tab.
  • Choose Probability Value and Left Tail for the region of the curve to shade.
  • Enter the Probability value as 0.02.
  • Click OK.

Output using the MINITAB software is given below:

b.

To determine

Find the probability of making a Type II error if μ=23.

c.

To determine

Find the probability of making a Type II error if μ=24.

d.

To determine

Find the probability of making a Type II error if μ=25.5.

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Chapter 9 Solutions

STATISTICS F/BUSINESS+ECONOMICS-TEXT
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